Origin of the mass splitting of azimuthal anisotropies in a multi-phase transport model

Origin of the mass splitting of azimuthal anisotropies in a multi-phase transport model

Hanlin Li College of Science, Wuhan University of Science and Technology, Wuhan, Hubei 430065, China Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA    Liang He Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA    Zi-Wei Lin Department of Physics, East Carolina University, Greenville, North Carolina 27858, USA Key Laboratory of Quarks and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan, Hubei 430079, China    Denes Molnar Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA    Fuqiang Wang fqwang@purdue.edu Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA School of Science, Huzhou University, Huzhou, Zhejiang 313000, China    Wei Xie Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA
July 18, 2019
Abstract

Both hydrodynamics-based models and a multi-phase transport (AMPT) model can reproduce the mass splitting of azimuthal anisotropy () at low transverse momentum () as observed in heavy ion collisions. In the AMPT model, however, is mainly generated by the parton escape mechanism, not by the hydrodynamic flow. In this study we provide detailed results on the mass splitting of in this transport model, including and of various hadron species in +Au and Au+Au collisions at the Relativistic Heavy Ion Collider and +Pb collisions at the Large Hadron Collider. We show that the mass splitting of hadron and in AMPT first arises from the kinematics in the quark coalescence hadronization process, and then, more dominantly, comes from hadronic rescatterings, even though the contribution from the latter to the overall charged hadron is small. We further show that there is no qualitative difference between heavy ion collisions and small-system collisions or between elliptic () and triangular () anisotropies. Our studies thus demonstrate that the mass splitting of and at low- is not a unique signature of hydrodynamic collective flow but can be the interplay of several physics effects.

pacs:
25.75.-q, 25.75.Ld

I Introduction

The quark-gluon plasma has been created in relativistic heavy ion collisions, and extensive efforts are going on to study quantum chromodynamics at the extreme conditions of high temperature and energy density Arsene:2004fa (); Back:2004je (); Adams:2005dq (); Adcox:2004mh (); Muller:2012zq (). Of particular interests are non-central heavy ion collisions, where the overlap volume of the colliding nuclei is anisotropic in the transverse plane perpendicular to the beam direction. One interesting finding is that the collision system is explosive, consistent with the buildup and expansion of the hydrodynamic pressure Heinz:2013th (); Gale:2013da (); Abelev:2008ab (). The pressure gradient and/or particle interactions would generate an anisotropic expansion, which converts the anisotropic geometry into the final-state elliptic flow Ollitrault:1992bk (). In addition, due to fluctuations in the initial-state collision geometry, there is an elliptic harmonic anisotropy in the configuration space ) even in central heavy ion or proton-nucleus collisions Andrade:2006yh (). Furthermore, fluctuations lead to finite configuration space harmonics of all orders Alver:2010gr (), which will result in final-state momentum anisotropies of all orders (), where is a positive integer.

The mass splitting of hadron at low transverse momentum () is also observed in the experimental data. It is often considered as a hallmark of the hydrodynamic description of relativistic heavy ion collisions Heinz:2013th (), where a common but anisotropic transverse velocity field coupled with the Cooper-Frye hadronization mechanism Cooper:1974mv () leads to the mass splitting. Furthermore, results from hybrid models, where hydrodynamics is followed by a hadron cascade, have shown that the mass splitting is small just after hadronization and is then strongly enhanced by hadronic scatterings Hirano:2007ei (); Song:2010aq (); Romatschke:2015dha ().

Large values have been observed in large-system heavy ion collisions, and both hydrodynamics-based models Heinz:2013th (); Gale:2013da (); Abelev:2008ab () and a multi-phase transport (AMPT) model Lin:2001zk (); Lin:2004en (); Lin:2014tya () can reproduce these results. Later particle correlation data in small systems, including +Au Adare:2014keg (); Adamczyk:2015xjc () collisions at the Relativistic Heavy Ion Collider (RHIC) and high multiplicity + Khachatryan:2010gv () or +Pb CMS:2012qk (); Abelev:2012ola (); Aad:2012gla () collisions at the Large Hadron Collider (LHC), hint at similar (and mass splitting). Again, both hydrodynamics Bozek:2010pb (); Bozek:2012gr () and a multi-phase transport Bzdak:2014dia () can reasonably describe the experimental data. This seems puzzling, because naively one would expect the small system to be far from equilibrium and thus not suitable for a hydrodynamical description.

A recent study by some of us He:2015hfa (); Lin:2015ucn () using AMPT has shown that the azimuthal anisotropy is mainly generated by the anisotropic parton escape and that hydrodynamics may play only a minor role. This escape mechanism would naturally explain the similar azimuthal anisotropies in heavy ion and small system collisions. Since mass splitting of hadron is also present in the AMPT results, it suggests that the hydrodynamic collective flow may not be the only mechanism that can generate the mass splitting of hadron in collisions with high energy densities.

In an earlier study Li:2016flp () we used AMPT simulations of Au+Au and +Au collisions at the top RHIC energy to investigate the mass splitting of of pions, kaons, and protons. We found that the mass splitting of in AMPT is partly due to the kinematics in the quark coalescence process but mainly due to hadronic rescatterings Li:2016flp (). In this paper we expand that study to more hadron species including , , , and strange hadrons such as and . We also investigate the massing splitting of the triangular flow and include AMPT results of +Pb collisions at the LHC energy of 5 TeV. In addition, we provide details of our analysis, such as the effect of the finite opening angles among coalescing partons, the difference between primordial hadrons and hadrons from resonance decays, and the connection between the mass splitting and the initial hadron spatial eccentricity.

Ii Model and Analysis

We employ the same version of the string melting AMPT model (v2.26t5, available online at AMPTcode ()) as in earlier studies  He:2015hfa (); Lin:2015ucn (); Li:2016flp (). It consists of a fluctuating initial condition, parton elastic scatterings, quark coalescence for hadronization, and hadronic interactions. The initial energy and particle productions are being described by the HIJING model. However, the string melting AMPT model converts these initial hadrons to their valence quarks and antiquarks, based on the assumption that the high energy density in the overlap region of high energy heavy ion collisions requires us to use parton degrees of freedom to describe the dense matter Lin:2001zk (). Two-body elastic parton scatterings are treated with Zhang’s Parton Cascade (ZPC) Zhang:1997ej (), where we take the strong coupling constant and a total parton scattering cross section  mb for all AMPT calculations in this study. After partons stop interacting, a simple quark coalescence model is applied to describe the hadronization process that converts partons into hadrons  Lin:2004en (). Subsequent interactions of these formed hadrons are modeled by a hadron cascade Lin:2004en ().

Two of the above components, the hadronization process and hadron cascade, are especially relevant for this study. Hadronization in the string melting version of AMPT is modeled with a simple quark coalescence, where two nearest partons in space (one quark and one antiquark) are combined into a meson and three nearest quarks (or antiquarks) are combined into a baryon (or antibaryon). In addition, when the flavor composition of the coalescing quark and antiquark allows the formation of either a pseudo-scalar or a vector meson, the meson species whose mass is closer to the invariant mass of the coalescing parton pair will be formed. The same criterion is also applied to the formation of an octet or a decuplet baryon with the same flavor composition. Thus in these situations the hadron species that has a larger mass will be typically formed when the coalescing partons have a larger invariant mass.

The hadron cascade in the AMPT model includes explicit particles such as , , , , , , mesons, , , , , , , , , and deuteron and the corresponding anti-particles Oh:2009gx (). Hadronic interactions include meson-meson, meson-baryon, and baryon-baryon elastic and inelastic scatterings. For example, meson-baryon scatterings includes pion-nucleon, -nucleon, and kaon-nucleon elastic and inelastic processes, among many reaction channels. More details can be found in the main AMPT paper Lin:2004en (). We terminate the hadronic interactions at a cutoff time (), when the observables of interest are stable; a default cutoff time of  fm/ is used here.

In this study we simulate three collision systems: Au+Au collisions at RHIC with -8.1 fm (corresponding to approximately 20%-30% centrality Abelev:2008ab ()) at the nucleon-nucleon center-of-mass energy  GeV, +Au collisions at RHIC with  fm at  GeV, and +Pb collisions at LHC with  fm at  TeV. Note that the string melting version of AMPT can reasonably reproduce the particle yields, spectra, and of low- pions and kaons in central and mid-central Au+Au collisions at 200A GeV and Pb+Pb collisions at 2760A GeV Lin:2014tya ().

The initial geometric anisotropy of the transverse overlap region of a heavy-ion collision is often described by the eccentricity of the th harmonic order Alver:2010gr ():

(1)

Here and are the polar coordinate of each initial parton (after its formation time) in the transverse plane, and denotes the per-event average. We follow the same method as in our earlier studies He:2015hfa (); Li:2016flp () to calculate azimuthal anisotropies. In particular, we compute the harmonic plane (short-axis direction of the corresponding harmonic component) of each event from its initial configuration of all partons Ollitrault:1993ba () according to

(2)

The momentum anisotropies are then characterized by Fourier coefficients Voloshin:1994mz ()

(3)

where is the azimuthal angle of the parton or hadron momentum. Note that all results shown in this paper are for particles (partons or hadrons) within the pseudo-rapidity window of .

Iii Partonic anisotropy

Currently the string melting version of the AMPT model  Lin:2001zk (); Lin:2004en (); AMPTcode () has only quarks but no gluons, where the gluon degree of freedom can be considered as being absorbed in the quark’s. Note that the scattering cross-sections in the parton cascade are set to be the same regardless of quark flavors. Figure 1 shows the and of the and light quarks and the strange quarks in three systems: Au+Au and +Au collisions at 200 GeV, and +Pb collisions at 5 TeV. The quark and antiquark anisotropies are found to be the same, so they are combined. There is practically no difference between the and quark ’s, so they are also combined in Fig. 1. The magnitudes are similar among the three systems, except in +Au which is significantly lower than the other two systems. In general small systems should generate lower than large systems, and this is the case for between +Au and Au+Au collisions. The in +Au is not much smaller than that in Au+Au, possibly because the lower energy density in +Au is compensated by the larger elliptical eccentricity (). The in +Pb are not much smaller than those in Au+Au, and this may be because the smaller system size is compensated by the larger collision energy.

Figure 1: (Color online) Parton . Parton (upper panels) and (lower panels) as a function of for light ( and ) and strange () (anti-)quarks in the final state before hadronization from AMPT with string melting. Three systems are shown: -8.1 fm Au+Au collisions at  GeV (left column),  fm +Au collisions at 200 GeV (middle column), and  fm +Pb at 5 TeV (right column). Both normal (solid curves) and -randomized (dashed curves) AMPT results are shown, with thick curves for light quarks and thin curves for strange quarks.

At low the light quark is larger than the quark’s. This is qualitatively consistent with the hydrodynamic picture where particles move with a common collective flow velocity. Because particles have the same at the same speed, as a function of are split according to particle masses. This mass splitting between light and strange quarks is observed in both and and in all three systems.

Since our previous studies He:2015hfa (); Lin:2015ucn () have shown that comes largely from the anisotropic escape mechanism, then the question is whether or not the observed mass splitting is entirely due to the minor contributions from hydrodynamics. So we also carry out a test calculation with no collective anisotropic flow by randomizing the outgoing parton azimuthal directions after each parton-parton scattering as in Ref. He:2015hfa (). The results are shown by the dashed curves in Fig. 1, where the differences between the light quark and strange quark ’s are still present. Since the parton azimuthal angles are now randomized, the final-state parton anisotropy is entirely due to the anisotropic escape mechanism He:2015hfa (). The fact that the mass splitting is similar between the normal and -randomized AMPT suggests that it is caused by the mass or kinematic difference in the scatterings rather than the collective flow. At high the light quark and strange quark ’s approach each other; this is expected because the mass difference becomes unimportant at high .

Iv Mass splitting from quark coalescence

Since there is mass splitting in the quark , it is natural to expect mass splitting in the of hadrons with different quark contents. However, for hadrons such as pions, -mesons, and protons made of light quarks only, the difference between their anisotropies must come from the hadronization process and/or hadronic rescatterings. We first study the effect of the former by examining of hadrons right after hadronization but before hadron rescatterings take place. Figure 2 shows the and of primordial , , , (), (), () as a function of in the three systems we studied. Note that primordial hadrons are hadrons formed directly from hadronization but before resonance decays and hadronic scatterings. In Au+Au collisions the particle exhibit the familiar mass-ordering at low : the ’s of pions are larger than those of kaons which are in turn larger than those of (anti-)protons and strange baryons. The mass splittings in the small systems of +Au and +Pb are not necessarily the same ordering as in the Au+Au system. In this section we study how this mass splitting comes about. We will concentrate on but the discussions can be extended to .

Figure 2: (Color online) Mass splitting from coalescence. Primordial hadron (upper panels) and (lower panels) as a function of right after quark coalescence but before hadronic rescatterings take place in AMPT with string melting. Three systems are shown: -8.1 fm Au+Au collisions at  GeV (left column),  fm +Au collisions at 200 GeV (middle column), and  fm +Pb at 5 TeV (right column). Thick solid curves are for charged pions, thin solid curves for charged kaons, thin dashed curves for -mesons, medium thick solid curves for (anti-)protons, thick dashed curves for (), and medium thick dashed curves for ().

Since the string melting version of AMPT forms hadrons via quark coalescence, the difference in the pion and proton comes from the difference in the number of constituent quarks and in the kinematics of those quarks. At high the hadron has been measured to exhibit the number of constituent quark (NCQ) scaling:

(4)

where the superscripts ‘B’ stands for baryons and ‘M’ for mesons. This comes naturally from quark coalescence, where two or three relatively high quarks are almost collimated and coalesce into a meson or baryon. The meson and baryon take on twice and three times the quark (which are saturated at high as in Fig. 1), respectively. This NCQ scaling is evident in Fig. 2; the baryons in each graph approach a similar magnitude of in the higher region.

However, this quark collimation picture cannot be extended to low , since there the relative momentum among constituent quarks could be comparable to the hadron , i.e. there will be finite opening angles among the constituent quarks. Therefore the kinematics in the coalescence process Lin:2009tk () such as finite opening angles could lead to the mass splitting of at low . To quantitatively understand this, we show in the upper panel of Fig. 3 the distributions for partons coalescing into pions and protons of GeV/. We have also depicted in the plots the meson, which has the same constituent quark content as the but a larger mass. The lower panel of Fig. 3 shows the absolute difference between the azimuthal angle of the constituent quark and that of the formed hadron, . Because of the finite angles, the average of the constituent quarks is larger than one half (one third) of the pion (proton) . While the actual kinematics are complex, one may verify that a pair (or triplet) of partons with an average transverse momentum at the average opening angle (as in Fig. 3) gives the composite hadron roughly as

(5)

where the superscripts ‘h’ and ‘q’ stand for hadrons and constituent quarks, respectively, and is the number of constituent quarks for the given hadron type.

Figure 3: (Color online) Coalescence kinematics. Final-state (upper panel) and azimuthal opening angle (lower panel) distributions of constituent (anti-)quarks forming , , and (anti-)proton. Shown are string melting AMPT results for 6.6-8.1 fm Au+Au collisions at 200 GeV.

Similarly, because of the finite opening angle the hadron is not simply twice (or three times) the average quark at the corresponding average quark . This is shown in Fig. 4, where the quark is plotted at the of the hadron it coalesces into, together with the hadron from Fig. 2. Note that the quark in Fig. 1 includes all quarks (i.e. from all hadrons) while that in Fig. 4 is categorized by the formed hadrons. As seen from Fig. 4, the hadron ’s shown in solid curves are smaller than twice (three times) the quarks shown in dashed curves. Note that the shapes of the quark curves are different from each other because they are plotted at the hadron and because samplings of quarks into pions, mesons, and protons are different (c.f. Fig. 3). One may get a semiquantitative understanding of the hadron curve by, again, using the average quark kinematics. The hadron azimuthal distribution is

(6)

Thus the hadron is given by

(7)

One may verify that this relationship, with the kinematics in Fig. 3, can approximately describe the relationship between a hadron at  GeV/ and its constituent quarks in Fig. 4.

Figure 4: (Color online) Conversion of constituent quark into hadron by coalescence. Primordial hadron and constituent quark , both plotted as a function of the hadron . The hadron is taken before any hadronic rescatterings and the constituent quark is taken just before coalescence. Shown are both normal (upper panel) and -randomized (lower panel) string melting AMPT results for -8.1 fm Au+Au collisions at 200 GeV.

Although is largely from the escape mechanism, there does exist a contribution from hydrodynamics in AMPT He:2015hfa (); Lin:2015ucn (). Thus we also carry out the test calculations with no collective anisotropic flow by randomizing the outgoing parton azimuthal directions after each parton-parton scattering as in Ref. He:2015hfa (); Lin:2015ucn (). The results are shown in the lower panel of Fig. 4. In the -randomized case, the final-state freezeout anisotropy is entirely due to the anisotropic escape mechanism. Since mass splitting is also observed in the randomized case, it indicates that the hydrodynamical collective flow is not required to generate the mass splitting in right after hadronization.

V Effects of resonance decays

What are shown in Fig. 4 are the values of primordial hadrons (obtained right after the quark coalescence in the AMPT evolution), not those of hadrons after resonance decays. Figure 5 shows the fraction of primordial pions, kaons, and protons as a function of . Since what we measure in detectors are particles after strong decays, we need to include the effects of resonances decays on . In this section, we thus set the maximum hadronic stage to  fm/ in AMPT (parameter NTMAX=3) which turns off hadronic rescatterings. We then obtain the final-state hadron that include decays. Note that the final freezeout particles in AMPT include all strong decays of resonances but no electromagnetic or weak decays by default (except for the decay in order to include its feed down to Lin:2004en ().

The left panel of Fig. 7 shows the of primordial pions, primordial ’s, pions from decays, and all pions. The middle panel shows the corresponding results for kaons where the decay channel is studied. The right panel shows the of primordial (anti-)protons, primordial (anti-)’s (as an example), protons from (anti-) decays (as an example), and all protons. We see that at low heavier primordial particles have smaller . In addition, the decay product is usually smaller than their parent . As a result, the ’s of final-state hadrons including the decay products are smaller than (or closely follow) those of the primordial particles. This reduction effect is stronger in pions than protons, because a bigger fraction of pions comes from resonance decays than protons according to Fig. 5 and because the protons retain more of the parent than pions due to kinematics.

Figure 5: (Color online) Decay contributions. Fraction of primordial hadrons in -8.1 fm Au+Au collisions at 200 GeV from string melting AMPT where hadronic rescatterings are turned off.
Figure 6: (Color online) Effect of decays on . The of primordial hadrons (thin solid curves), resonances (thin dashed curves), decay products (thick dashed curves), and the total including both the primordial hadrons and decay products (thick solid curves). Shown are -8.1 fm Au+Au collisions at 200 GeV from string melting AMPT where hadronic scatterings are turned off.
Figure 7: (Color online) Effect of decays on mass splitting. Hadron (upper panels) and (lower panels) including both primordial hadrons and decay products. Three systems are shown: -8.1 fm Au+Au collisions at  GeV (left column),  fm +Au collisions at 200 GeV (middle column), and  fm +Pb at 5 TeV (right column). Results are from string melting AMPT where hadronic rescatterings are turned off. Thick solid curves are for charged pions, thin solid curves for charged kaons, medium thick solid curves for (anti-)protons, thick dashed curves for (), and medium thick dashed curves for ().
Figure 6: (Color online) Effect of decays on . The of primordial hadrons (thin solid curves), resonances (thin dashed curves), decay products (thick dashed curves), and the total including both the primordial hadrons and decay products (thick solid curves). Shown are -8.1 fm Au+Au collisions at 200 GeV from string melting AMPT where hadronic scatterings are turned off.

Our results generally agree with those in Refs. Eyyubova:2009hh (); Qiu:2012tm (); Crkovska:2016flo (). It is interesting to note that the -decay pion curve at low does not follow the trend of the parent -meson . Since the decay pion momentum in the rest frame is about 0.36 GeV/, in order to have a low- daughter pion in the lab frame, the decay must be very asymmetric: one pion at low and the other at high . The high- pion closely follows the parent direction, while the low- pion aligns more perpendicularly due to momentum conservation. With positive of the , there are therefore relatively more low- decay pions perpendicular to the reaction plane, hence a negative pion . We have verified that this feature is also true for the pions from decays.

Figure 7 shows the hadron and as a function of including contributions from resonance decays. The reduction in is evident in Fig. 7 in comparison with Fig. 2 where only the primordial hadron ’s are shown. Because of the larger reduction in pion than in proton due to decays, the amount of mass-splitting is reduced. Depending on the magnitude of this reduction, the mass splitting between primordial hadrons right after coalescence may or may not survive once including the decay products. So in general the mass splitting effect decreases after including the decay products.

Vi Mass splitting from hadronic rescatterings

Another source of mass splitting of comes from hadronic rescatterings. In the following we study as a function of the degree of hadronic rescatterings. We achieve this by varying the maximum allowed time, , of the hadronic interaction stage in AMPT. So can be considered as a qualitative indicator of the amount of hadronic rescatterings. Note that there is no cut-off time for the partonic evolution in AMPT.

The upper panels of Fig. 8 show the of charged pions, charged kaons, (anti)protons and charged hadrons (here defined as the sum of charged pions, kaons, protons and antiprotons) at freezeout in mid-central Au+Au collisions versus for various values. We see that the pion increases with the amount of rescatterings, the proton decreases, while the kaon does not change significantly. This can be understood as the consequence of hadron interactions. For example, pions and protons tend to flow together at the same velocity due to their interactions. Thus, pions and protons at the same velocity (i.e. small pions and large protons) will tend to have the same anisotropy after rescatterings. This will then lead to lower for protons and higher for pions at the same value. Similar conclusions were reached in previous hadron cascade studies Burau:2004ev (); Petersen:2006vm (); Zhou:2015iba () and a recent study with free-streaming evolution coupled to a hadron cascade Romatschke:2015dha ().

Figure 8: (Color online) Effects of hadronic rescatterings on . Hadron as a function of at different stages of hadronic rescatterings in string melting AMPT. The are for final-state hadrons including resonance decays, where the final freezeout is controlled by the maximum allowed time () for the hadronic stage. Shown are the results for charged pions (first column), charged kaons (second column), (anti-)protons (third column), and charged hadrons (last column) in -8.1 fm Au+Au collisions at 200 GeV (upper panels) and  fm +Au collisions at 200 GeV (lower panels).

Figure 8 also shows a small increase in the overall charged hadron , and this is due to the remaining finite configuration space eccentricity before hadronic scatterings take place. In general, whether there is an overall gain in the of charged hadrons depends on the configuration geometry at the beginning of hadron cascade. The lower panels of Fig. 8 shows our results for +Au collisions. We see that the pion increases significantly with hadronic scatterings while the proton remains roughly unchanged. Note that the overall gain in the charged hadron is larger in +Au than Au+Au collisions, and this is due to the larger eccentricity in the +Au system at the start of hadron cascade. Therefore the changes in the pion and proton are a net effect of the mass splitting due to pion-proton interactions (i.e. increase in the pion and decrease in the proton ) and the overall gain of for charged hadrons.

As can be seen in Fig. 8, continues to develop after hadronization in Au+Au as well as +Au collisions. In Au+Au collisions the development happens mainly during 5-20 fm/  while in +Au collisions the development happens earlier (mainly before 5 fm/). The spatial anisotropy is self-quenched due to the expansion and the development of momentum space anisotropy. The further increase of overall charged hadron in Fig. 8 suggests that the spatial anisotropy is not completely quenched at the time right after hadronization; a finite spatial anisotropy is present at the beginning of hadronic rescatterings which results in the further development of .

We elaborate this further by examining the increase as a function of the remaining eccentricity after hadronization (), i.e. the starting eccentricity for hadronic cascade. This is shown in Fig. 9 for both Au+Au and +Au collisions. Since a typical AMPT evolution around mid-rapidity essentially ends by the time of 30 fm/, we evaluate the increase in from hadronic scatterings as . The value is calculated with respect to the initial configuration space , as is . We have verified that the hadron right after the coalescence hadronization, as well as the at final freezeout, is proportional to the initial eccentricity ()–which is also calculated with respect to the initial –except when is large (close to one). We have also found that the value is positively correlated with the value in Au+Au collisions, while the correlation is weak in +Au collisions.

Figure 9: (Color online) Connection to the initial hadronic eccentricity. Gain in charged hadron due to hadronic rescatterings from hadronization ( fm/) to final freezeout ( fm/) as a function of the configuration space eccentricity of hadrons right after hadronization () in -8.1 fm Au+Au (circles) and  fm +Au (triangles) collisions at 200 GeV. The are for final-state hadrons including resonance decays.

Figure 9 show that, in the range of 0-0.2 in Au+Au and 0-0.5 in +Au collisions, roughly increases linearly with . At large positive , the statistics are poor and may not reflect a bulk geometry any more. At negative events are also rare. On average, is 0.11 in Au+Au and 0.42 in +Au collisions, starting from an initial of 0.29 and 0.53 for the two collision systems, respectively. The geometric anisotropy is thus not quenched completely after partonic interactions in Au+Au collisions; the reduction in eccentricity in +Au collisions is even smaller due to a shorter partonic stage. The remaining spatial anisotropy is smaller in Au+Au than in +Au collisions, and this results in a smaller gain during the hadronic rescattering stage in Au+Au than in +Au collisions, as observed in Fig. 8.

It is also interesting to note in Fig. 9 that is finite for events with , where one would naively expect . This would indeed be true if the initial hadron (before hadronic rescatterings) was zero, analogous to the zero initial parton anisotropies in AMPT (before partonic scatterings). However, for finite initial , which is the case here, it is not necessarily true that would not further develop.

Figure 10 shows hadron as a function of before hadronic rescatterings but including resonance decays in dashed curves and of freezeout hadrons after hadronic rescatterings in solid curves. As shown, hadronic rescatterings make significant contributions to the mass splitting in the final-state hadron . Meanwhile the absolute gain of the magnitude is relatively small during the hadronic stage.

Figure 10: (Color online) Effect of hadronic rescatterings on mass splitting. Charged pions (thick curves), charged kaons (thin curves), and (anti-)proton (medium thin curves) as a function of before (dashed curves) and after (solid curves) hadron rescatterings in string melting AMPT. The effects of resonance decays are included. Three systems are shown: -8.1 fm Au+Au collisions at  GeV (left column),  fm +Au collisions at 200 GeV (middle column), and  fm +Pb at 5 TeV (right column).
Figure 11: (Color online) Mass splitting at freezeout. Final hadron (upper panels) and (lower panels) as a function of from string melting AMPT, where resonance decays are included. Three systems are shown: -8.1 fm Au+Au collisions at  GeV (left column),  fm +Au collisions at 200 GeV (middle column), and  fm +Pb at 5 TeV (right column). Thick solid curves are for charged pions, thin solid curves for charged kaons, thin dashed curves for -mesons, medium thick solid curves for (anti-)protons, thick dashed curves for (), and medium thick dashed curves for ().

Vii Discussions

Figure 11 shows and of final-state hadrons as a function of in Au+Au and +Au collisions at 200 GeV and +Pb collisions at 5 TeV. The -mesons are all decayed in the final state of AMPT, so they are reconstructed by the invariant mass of pairs Pal:2002aw () and pairs with combinatorial background subtraction, as usually done in experiments Abelev:2008aa (). The mass splitting of at low is more obvious in Au+Au collisions than small systems. There is also splitting in at high , likely more due to the number of constituent quarks rather than the mass difference.

We summarize our results on the mass splitting of with Fig. 12, which shows the of pions, kaons, protons and anti-protons, and charged hadrons within a fixed bin of  GeV/, as an example. Different stages of the collision system evolution are shown: (i) right after the quark coalescence hadronization including only primordial particles (data points plotted to the left of ); (ii) right after the quark coalescence but including resonance decays (data points plotted at  fm/); (iii) after various degrees of hadronic rescatterings, which are obtained from freezeout particles by setting to the corresponding values as plotted. As shown in Fig. 12, most of the overall is built up in the partonic phase, while the additional gain in the overall from hadronic rescatterings is small. On the other hand, although there is often a significant mass splitting in the primordial hadron right after hadronization due to the kinematics in the quark coalescence process, the mass splitting is often reduced when decay products are included in . In other words, the mass splitting before hadronic rescatterings is usually small. This small mass splitting does not change significantly during the first 5 fm/ in Au+Au collisions, since the partonic stage dominates the early evolution. We also see that a significant mass splitting is built up during the time of 5-20 fm/ of hadronic rescatterings. After 20 fm/ there is little further change in the in +Au or +Pb collisions, while in Au+Au there is still a small increase in the size of mass splitting.

Figure 12: (Color online) Origin of mass splitting. The (upper panels) and (lower panels) of charged pions, charged kaons, (anti-)protons, and charged hadrons within  GeV/ at different stages of the collision evolution. Points plotted within the shaded areas represent primordial hadrons right after quark coalescence, points plotted at  fm/ represent results right after hadronization but including resonance decays, and points at various values represent results after hadronic rescatterings (and resonance decays).

Viii Conclusions

We have studied the developments of the mass splitting of hadron at different stages of nuclear collisions with a multi-phase transport model AMPT. First results on Au+Au and +Au collisions at the top RHIC energy have been published in Ref. Li:2016flp (). The present work provides extensive details to that earlier study by including more hadron species such as resonances and strange hadrons. We also expand the investigation to the triangular flow and +Pb collisions at the LHC energy of 5 TeV. We reach the same conclusion for and , for both heavy ion collisions and small-system collisions, in that the mass splitting of hadron is partly due to the quark coalescence hadronization process but more importantly due to hadronic rescatterings. Although the overall amplitude is dominantly developed during the partonic stage, the mass splitting is usually small right after hadronization, especially after including resonance decays. The majority of the hadron mass splitting is developed in the hadronic rescattering stage, even though the gain in the overall of charged particles is small there. These qualitative conclusions are the same as those from hybrid models that couple hydrodynamics to a hadron cascade, even though in transport models such as AMPT the anisotropic parton escape is the major source of . In the -randomized test of AMPT, where the anisotropic parton escape is the only source of , we also observe similar mass splitting of hadron . Therefore we conclude that the mass splitting of can be an interplay of several physics processes and is not a unique signature of hydrodynamic collective flow.

Acknowledgments

This work is supported in part by US Department of Energy Grant No. DE-SC0012910 (LH,FW,WX), No. DE-FG02-13ER16413 (DM), and the National Natural Science Foundation of China Grant No. 11628508 (ZWL) and No. 11647306 (FW). HL acknowledges financial support from the China Scholarship Council.

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